The 1 W resistor will get less hot that the 1/4 W resistor if they both dissipate the same power. The specific heat may be comparable, but because of the higher mass the 1 W resistor will need more power to get the same temperature rise.
You may need to place several resistors in series to prevent overheating. suppose you have a 1 kΩ/ 1/4 W resistor which has 20 V across it. Then the power will be (20 V)\$^2\$/ 1 kΩ = 400 mW, which is more than the 1/4 W the resistor is rated for, and which will reduce the resistor's life. You can use a 1 W version instead or for instance three 330 Ω/ 1/4 W resistors in series. Each will then dissipate only 130 mW, so that's a safe value.
Note that resistors can only dissipate their rated power at low temperatures. Most must be derated above 70 °C environment temperature, which means that the higher you go beyond that temperature the less power it may dissipate, until its maximum temperature, where the allowed dissipation becomes zero.
Apart from spreading power you also may need a couple of series resistors for high voltage applications. A resistor may be rated at 160 V, then you can't use it for 230 V, even if the current (and thus power) are very low. 230 V AC is 325 V peak, so you'll need 3 resistors in series.